Abstract
Given a metric space X and a subspace \(A\subset X\), we prove that A can generate various algebraic elements in persistent homology of X. We call such elements (algebraic) footprints of A. Our results imply that footprints typically appear in dimensions above \(\dim (A)\). Higher dimensional persistent homology thus encodes lower dimensional geometric features of X. We pay special attention to a specific type of geodesics in a geodesic surface X called geodesic circles. We explain how they may generate non-trivial odd-dimensional and two-dimensional footprints. In particular, we can detect even some contractible geodesics using two- and three-dimensional persistent homology. This provides a link between persistent homology and the length spectrum in Riemannian geometry.
Similar content being viewed by others
References
Adamaszek, M., Adams, H.: The Vietoris-Rips complexes of a circle. Pac. J. Math. 290-1, 1–40 (2017)
Adamaszek, M., Adams, H., Reddy, S.: On Vietoris-Rips complexes of ellipses. J. Topol. Anal. 11, 661–690 (2019)
Adams, H., Chowdhury, S., Jaffe, A., Sibanda, B.: Vietoris-Rips complexes of regular polygons. arXiv:1807.10971
Adams, H., Coldren, E., Willmot, S.: The persistent homology of cyclic graphs. arXiv:1812.03374
Adams, H., Coskunuzer, B.: Geometric Approaches on Persistent Homology. arXiv:2103.06408
Attali, D., Lieutier, A., Salinas, D.: Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes. In: Proceedings of the 27th annual ACM symposium on Computational geometry, SoCG ’11, pp 491–500, New York, NY (2011)
Bauer, U.: Ripser. https://github.com/Ripser/ripser (2006)
Cencelj, M., Dydak, J., Vavpetič, A., Virk, Ž: A combinatorial approach to coarse geometry. Topol. Appl. 159, 646–658 (2012)
Chambers, E.W., de Silva, V., Erickson, J., Ghrist, R.: Rips complexes of planar point sets. Discr. Comput. Geom. 44(1), 75–90 (2010)
Chambers, E.W., Letscher, D.: On the height of a homotopy. In: Proceedings of the 21st Canadian Conference on Computational Geometry, pp. 103–106 (2009)
Chazal, F., Crawley-Boevey, W., de Silva, V.: The observable structure of persistence modules. Homol. Homotop. Appl. 18(2), 247–265 (2016)
Chazal, F., de Silva, V., Oudot, S.: Persistence stability for geometric complexes. Geom. Dedicat. 173, 193 (2014)
Čufar, M.: Računanje enodimenzionalne vztrajne homologije v geodezični metriki. Ms Thesis, University of Ljubljana (2020)
Dranishnikov, A.: Anti-Čech approximation in coarse geometry. Preprint, Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France (2002)
Dydak, J., Segal, J.: Shape Theory. An Introduction. Springer, Berlin (1978)
Edelsbrunner, H., Wagner, H.: Topological data analysis with Bregman divergences. In: Proc. 33rd Ann. Sympos. Comput. Geom. 39:1–39:16 (2017)
Frosini, P.: Metric homotopies. Atti del Seminario Matematico e Fisico dell’Università di Modena, XLVII, 271–292 (1999)
Gasparovic, E., Gommel, M., Purvine, E., Sazdanovic, R., Wang, B., Wang, Y., Ziegelmeier, L.: A Complete Characterization of the \(1\)-Dimensional Intrinsic Čech Persistence Diagrams for Metric Graphs, In: Chambers E., Fasy B., Ziegelmeier L. (eds) Research in Computational Topology. Association for Women in Mathematics Series, vol 13. Springer, Cham
Gornet, R., Mast, M.B.: The length spectrum of Riemannian two-step nilmanifolds. Annales scientifiques de l’École Normale Supérieure, Serie 4, Volume 33 no. 2, pp. 181–209 (2000)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Hausmann, J.-C.: On the Vietoris-Rips complexes and a cohomology theory for metric spaces. Ann. Math. Stud. 138, 175–188 (1995)
Hopf, H.: Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104, 637–665 (1931)
Lablée, O.: Spectral Theory in Riemannian Geometry. European Mathematical Society (2015)
Latschev, J.: Rips complexes of metric spaces near a closed Riemannian manifold. Arch. Math. 77(6), 522–528 (2001)
Lim, S., Memoli, F., Okutan, O.B.: Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius. arXiv:2001.07588
Niyogi, P., Smale, S., Weinberger, S.: Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39, 419–441 (2008)
Virk, Ž: 1-Dimensional intrinsic persistence of geodesic spaces. J. Topol. Anal. 12, 169–207 (2020)
Virk, Ž: Approximations of \(1\)-dimensional intrinsic persistence of geodesic spaces and their stability. Rev. Mat. Complutense 32, 195–213 (2019)
Virk, Ž: Rips complexes as nerves and a functorial Dowker–Nerve diagram. Mediterr. J. Math. 18, 58 (2021)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Research was supported by Slovenian Research Agency under Grant No. N1-0114 and P1-0292. The author would like to thank the referee for a careful reading and useful comments.
Rights and permissions
About this article
Cite this article
Virk, Ž. Footprints of Geodesics in Persistent Homology. Mediterr. J. Math. 19, 160 (2022). https://doi.org/10.1007/s00009-022-02089-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-022-02089-0